Superluminal apparent motions in distant radio sources

We derive the prediction of the standard model of superluminal radio sources for the apparent transverse velocity of a radio source located at redshift z. The apparent velocity of the source is reduced by a factor of 1 + z compared to that of a similar nearby source. The cause of this reduction is recession of the distant source due to the expansion of the universe. The apparent velocity of a source can be estimated from its redshift and proper motion using the values of the Hubble constant and the mean densities of different energy components in the universe. We derive an expression for the velocity valid for the currently favored cosmological model: a flat universe with a nonzero cosmological constant. (c) 2005 American Association of Physics Teachers.